ADM Decomposition of Metric

Main

(1)
\begin{align} ds^{2} = \left( -N^{2} + N_{i} N^{i} \right) dt^{2} + 2 N_{i} dt dx^{i} + q_{ij} dx^{i} dx^{j} \end{align}
(2)
\begin{align} I = \int \left( \pi^{ij} \left( \frac{\partial q_{ij}}{ \partial t} \right) - NH - N_{i} H^{i} \right) d^{4}x \end{align}

Homogeneous models with trivial momentum constraints, like Bianchi IX:

(3)
\begin{align} I = \int \left( \pi^{ij} \left( \frac{\partial q_{ij}}{ \partial t} \right) - NH \right) dt \end{align}
(4)
\begin{equation} ds^{2} = -N^{2} dt^{2} + q_{ij} dx^{i} dx^{j}. \end{equation}

Hamiltonian constraint:

(5)
\begin{align} H = G_{ij,kl} \pi^{ij} \pi^{kl} - q^{1/2} \left( ^{3}R - 2 \Lambda \right) = 0, \end{align}

where $G_{ij,kl} = \frac{1}{2} q^{1/2} \left( q_{ik} q_{jl} + q_{il} q_{jk} - q_{ij} q_{kl} \right)$ and $^{3} R = q^{ij} \left( \Gamma ^{k} _{ij,k}- \Gamma ^{k} _{ik,j} + \Gamma ^{k} _{ij} \Gamma ^{l} _{kl} - \Gamma ^{l} _{ik} \Gamma ^{k} _{jl} \right)$

FRW metric:

(6)
\begin{align} ds^{2} = -N^{2} \left( t \right) dt^{2} + a^{2} \left( t \right) \left( d \theta_{1}^{2} + \sin^{2} \theta_{1} \left( d \theta_{2}^{2} + \sin^{2} \theta_{2} d \phi^{2} \right) \right) \end{align}
(7)
\begin{align} \Rightarrow H = 12 \pi^{2} \left[ - \frac{a(t)}{N^{2}} \dot a^{2}(t) - a(t) + \frac{\Lambda}{3}a^{3}(t) \right] \end{align}

From Einstein action,

(8)
\begin{align} L = 12 \pi^{2} N \left( - \frac{a \dot a ^{2}}{N^{2}} + a - \frac{\Lambda}{3}a^{3} \right) \end{align}
(9)
\begin{align} \Rightarrow p = \frac{ \partial L}{\partial \dot a} = - \frac{24 \pi^{2}}{N} a \dot a \end{align}
(10)
\begin{align} \Rightarrow H = - \frac{1}{48 \pi^{2}} \frac{p^{2}}{a} + 12 \pi^2 \left( -a + \frac{\Lambda}{3} a^{3} \right) \end{align}
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